Hadronic bound-states in SU(2) from Dyson--Schwinger Equations
EEPJ manuscript No. (will be inserted by the editor)
Hadronic bound-states in SU(2) from Dyson–SchwingerEquations
Milan Vujinovic and Richard Williams Institut f¨ur Physik, Karl-Franzens–Universit¨at Graz, Universit¨atsplatz 5, 8010 Graz, Austria. Institut f¨ur Theoretische Physik, Justus-Liebig–Universit¨at Giessen, 35392 Giessen, Germany.Received: date / Revised version: date
Abstract.
By using the Dyson-Schwinger/Bethe-Salpeter formalism in Euclidean spacetime, we calculatethe ground state spectrum of J ≤ PACS.
Quantum Chromodynamics (QCD) is a strongly interact-ing gauge theory whose study has proven to be one of themost formidable challenges of modern theoretical physics.Whilst the high-energy regime of QCD is by now relativelywell explored in terms of perturbation theory, the arguablymore interesting (and intrinsically non-perturbative) phe-nomena such as dynamical chiral symmetry breaking andconfinement are yet to be fully understood.One of the strategies which might lead to our betterunderstanding of QCD is to investigate theories which areQCD-like, but have certain properties that make themtechnically less challenging than QCD itself. A prime ex-ample is provided by studies of SU(2) gauge theories withan even number of fermion flavors. Lattice simulations ofthese theories at non-zero chemical potential do not suf-fer from the sign problem, and such models thus provideideal conditions to study the phase diagram of stronglyinteracting matter [1–11].Here we wish to concentrate on the situation with twofundamentally charged Dirac fermions [1, 8, 10, 11]. Such atheory may also be interesting in the context of a unifieddescription of cold asymmetric Dark Matter (DM) anddynamical electroweak (EW) symmetry breaking [12–14],wherein the ground state hadronic spectrum at T = 0, µ = 0 is of great importance. It is exactly this hadronicspectrum that will be the central focus of our study.In this paper we use the non-perturbative, continu-ous and covariant formalism of Dyson-Schwinger (DSE) a e-mail: [email protected] b e-mail: [email protected] and Bethe-Salpeter (BSE) equations in Euclidean space-time [15–18]. When applied to QCD, the most commontruncation one can make is that of rainbow-ladder (RL),wherein the quark-antiquark interaction kernel is replacedby a dressed one gluon exchange. It is the simplest ap-proximation scheme that respects the axial-vector Ward-Takahashi identity (axWTI), thus preserving the chiralproperties of the theory and the (pseudo)-Goldstone bo-son nature of light pseudoscalar mesons. With a judiciouschoice of model dressing functions, the RL truncation hasbeen applied relatively successfully to QCD phenomenol-ogy for both mesons [19–30] and baryons [31–35].However, as we will show in this paper, the RL trunca-tion performs unsatisfactorily when adapted to an SU(2)theory with 2 fundamental flavors, even though the the-ory is expected to have QCD-like dynamics. We discusspossible reasons for this in more detail in Section 2. Herewe only comment that we strongly believe that (most) ofthe inadequacy of RL method comes from its weak con-nection to the underlying gauge sector. Remedying thisrequires the use of beyond rainbow ladder (BRL) tech-niques, with our preference towards those based on the di-agrammatic expansion of quark-gluon vertex DSE [36–46].Whilst there are other BRL methods available [47–53], wechoose the diagrammatic approach as it makes it easier tostudy the influence of the gauge sector on hadronic observ-ables. Our aim in this paper is thus not only to provide acontinuum calculation complimentary to the lattice inves-tigations of [12, 13], but also to explicitly demonstrate theimportance of using BRL methods when studying genericstrongly interacting theories. a r X i v : . [ h e p - ph ] A p r Milan Vujinovic , Richard Williams : Hadronic bound-states in SU(2) from Dyson–Schwinger Equations This manuscript is organised as follows. In Section 2we discuss the DSEs relevant for our calculation, and alsodescribe in some detail the approximations and model in-puts we employ. In Section 3 we describe the extrapolationprocedures used to obtain hadron masses, and provide es-timates for errors coming from extrapolation. The resultsare disscused and compared to relevant lattice data in Sec-tion 4. We conclude in Section 5.
In a theory with 2 colors, both mesons and baryons (di-quarks) can be described in terms of a two-body Bethe-Salpeter equation. For the meson[ Γ M ( p, P )] ij = (cid:90) k [ K ( p, k, P )] ik ; lj [ χ M ( k, P )] kl , (1)where (cid:82) k stands for (cid:82) d k/ (2 π ) and Γ M ( p, P ) is the me-son amplitude with appropriate J P C quantum numbers,relative momentum p and total momentum P , and themeson wavefunction is χ M ( k, P ) = S ( k + ) Γ M ( k, P ) S ( k − ).The quark propagators are S ( k ± ), at momenta k + = k + ηP and k − = k − (1 − η ) P , with k the loop momentum and η ∈ [0 ,
1] the momentum partition factor. In a covariantstudy, the results are independent of η : for concreteness,we work with η = 1 /
2. The final ingredient in Eq. (1) isthe quark-antiquark 4-point interaction kernel K ( p, k, P ).A diagrammatic representation of Eq. (1) for mesons isgiven in Fig. 1.In order to solve the BSE, one clearly needs as inputthe quark propagator S ( p ). This Green’s function is de-composed as S − ( p ) = Z − f ( p ) (cid:2) i/p + M ( p ) (cid:3) , (2)with Z f ( p ) the quark wavefunction and M ( p ) the dy-namical quark mass. The tree-level form is given by S − ( p ) = i/p + Z m m , where Z m is the quark mass renormalisationconstant. The quark propagator satisfies its own DSE, seeFig. 2, and is given by S − ( p ) = Z S − ( p ) (3)+ g Z f C F (cid:90) k γ µ S ( k + p ) Γ ν ( k + p, p ) D µν ( k ) . Here, Γ ν ( p, k ) and D µν ( k ) are the full quark-gluon vertexand gluon propagator, respectively. Renormalisation con-stants of the quark field and quark-gluon vertex are Z and Fig. 1.
The Bethe–Salepter equation for the meson.
Fig. 2.
The Dyson–Schwinger equation for the quark propaga-tor. Straight lines are quarks, wiggly ones gluons. Filled circlesindicate dressed propagators and vertices. Z f . They are related through a Slavnov-Taylor identitywhich takes a simple form when employing a miniMOMscheme [54] in Landau gauge, Z f = Z / (cid:101) Z with (cid:101) Z therenormalisation of the ghost propagator.The 4-point interaction kernel K ( p, k, P ) of Eq. (1) isconnected to the self-energy part Σ ( p ) of quark propaga-tor DSE through the axial-vector Ward-Takahashi identity(axWTI)[ Σ ( p + ) γ + γ Σ ( p − )] ij = (4) (cid:90) k [ K ( p, k, P )] ik ; lj [ Σ ( k + ) γ + γ Σ ( k − )] kl . This identity encodes the chiral properties of the theory,and severely constrains the form of the BSE interactionkernel once an approximation for the quark DSE has beenchosen. A direct connection is provided through the actionof ‘cutting’ internal quark lines [36, 37].
The ‘rainbow’ part of RL truncation refers to the replace-ment of the full quark-gluon vertex in Eq. (3) by Γ ν ( k + p, p ) → λ ( k ) γ ν , (5)i.e. its tree-level form augmented by a model dressing func-tion, λ ( k ), that is a function of the gluon momentumalone. The corresponding axWTI-preserving approxima-tion for BSE kernel is that of one gluon exchange (the‘ladder’), which we show diagrammatically in Fig. 3.In the RL approach, the model dressing function λ ( k )of Eq. (5) is often combined with the dressing of thegluon propagator D µν ( k ) into a single model function,constructed to reproduce correctly some hadronic observ-ables, usually m π and f π . Whilst this method has shownconsiderable success in QCD phenomenology (see e.g. [55,56] for some of the limitations of the model), in an SU(2)theory the approach seems rather unsuitable, especially inthe 1 ++ channel: see Table 2 for details. Fig. 3.
The truncated two-body kernel in rainbow-ladder ap-proximation.ilan Vujinovic , Richard Williams : Hadronic bound-states in SU(2) from Dyson–Schwinger Equations 3 There are two primary reasons why RL will not per-form satisfactorily in a generic strongly-interacting theory.One reason is with regards to its very limited interactionstructure ( γ ν × γ µ ) which offers no variation in interac-tion strength across different meson channels. The secondis that the connection to the underlying gauge dynamicsis typically lost in the construction of an effective quark-gluon interaction; this prevents adequate rescaling of pa-rameters such as g N c that cannot be translated into are-parameterization of an effective model. A BRL approach which is well suited for studying the in-fluence of underlying Yang-Mills sector on the hadronicobservables is based on the quark-gluon vertex [41, 44,46, 57–62]. Here, we focus on the truncated form of theDSE [46] shown in Fig. 4. Within this approximation, onlythe so-called non-Abelian (NA) diagram is kept in thequark-gluon vertex self-energy. The truncated kernel, con-sistent with constraints from chiral symmetry, is shown inFig. 5.So that the Bethe-Salpeter equation can be tackled,the evaluation of a fully self-consistent quark-gluon ver-tex is not performed. That is, the full calculated vertex(denoted by a red filled circle in Fig. 4) is not back-coupled into the non-Abelian diagram. Instead, the in-ternal vertices (orange squares in Fig. 4) are modelled bythe Eq. (5) with λ ( k ) constructed such that it stronglyresembles the tree-level projection of the full quark-gluonvertex at each iteration step; essentially, it depends upona function Λ ( M ) that encodes the interaction strength interms of the dynamically generated quark mass. We usedthe parametrisation Eq. (21) of Ref. [46], with modifica-tions that account for the change N c = 2 and the rescal-ing of the gauge coupling g ( g N c is left invariant). For Λ ( M ) we use the functional form given in Eq. (22) of [46]with parameters a (cid:39) . , b (cid:39) . , c (cid:39) − . , d (cid:39) . λ ( k ) of Eq. (5) follows from the tree-level projection of Fig. 4.
The truncated DSE for the quark-gluon vertex. Theorange square denotes the internal QG vertex model, accordingto Eq. (5).
Fig. 5.
The truncated two-body kernel beyond rainbow-ladderapproximation. the full quark-gluon vertex. We will re-iterate this point inSection 4.1, when we provide an estimation of the modeldependence.The final ingredient which we need to specify in ourcalculation is the gluon propagator D µν ( k ). We work inLandau gauge, where this Green’s function takes the form D µν ( k ) = T µν ( k ) Z ( k ) k , (6)with T µν ( k ) = δ µν − k µ k ν /k the transverse projectorwith respect to momentum k . The gluon dressing func-tion which we use is plotted in Fig. 6. The details of thisfunction and its parametrisation can be found in [63]. Wepoint out that the gluon which we employ corresponds toa quenched DSE calculation. Ignoring the back-reactionof quarks onto the Yang-Mills sector is usually consid-ered a good approximation for theories with QCD-like dy-namics, as the corresponding effect on ‘observables’ likethe chiral condensate, f π and others is quite small [64].However, the quenched approximation should be reconsid-ered in theories which have (nearly) conformal, or ‘walk-ing’ dynamics. Walking dynamics arises naturally in mod- -4 -3 -2 -1 -4 -3 -2 -1 p [arbitrary units] Z ( p ) G ( p ) Fig. 6.
Ghost ( G ) and gluon ( Z ) dressing functions employedin our calculations. The momentum p is in arbitrary units:scale setting procedure is described in Section 4.2. Milan Vujinovic , Richard Williams : Hadronic bound-states in SU(2) from Dyson–Schwinger Equations els with a relatively large number of light fundamentallycharged fermions [65–73], or fermions belonging to higher-dimensional representations of the gauge group [74–82]. P One of the consequences of working with Euclidean space-time is that access to time-like quantities, such as massesof bound-states, requires an analytic continuation of thecomponent Green’s functions to complex momenta. Whilstthis is only a minor technicality thanks to many well-established techniques in the literature [22, 49, 83–85],there are situations in which existing methods do not ap-ply, or which are simply too complicated to implement. Inthis case, indirect methods can be employed that enableaccess to a limited number of time-like quantities [86, 87].In the next two sections we describe two techniquesthat have been widely used, and compare their perfor-mance in cases where direct analytic continuation is possi-ble. This provides an estimate of the methods applicabilityto the study at hand.
There are several means by which the mass spectrum ofthe BSE can be obtained. The most often used is throughsolution of Eq. (1), written as a matrix equation for sim-plicity Γ i = λ (cid:0) P (cid:1) K ij Γ j . (7)This has solutions at discrete values of the bound-state’stotal momentum P = − M i . By introducing the func-tion λ ( P ) on the right, we obtain an eigenvalue equationwhose bound-state solution correspond to λ (cid:0) P (cid:1) = 1.Since λ ( P ) is a continuous function of P , one canconceive that its continuation from spacelike P > P < g ( λ ) = 1 − /λ , see Ref. [26], removes a con-siderable degree of intrinsic curvature in the region closeto the pole, rendering simple linear extrapolation viableprovided the extrapolation is not far .In the top panel of Fig. 7 we show the eigenvalue ex-trapolation of λ (cid:0) P (cid:1) for various J P C states. The data isfirst transformed via g ( λ ), before a linear fit f (cid:0) P (cid:1) = a + bx is performed. Finally, we plot the inverse functionof g , λ fit = g − (cid:0) f (cid:0) P (cid:1)(cid:1) as solid lines. Exact results, ob-tained via calculation in the complex plane, are includedas labelled points. The second means to obtain the mass spectrum employsinstead the inhomogeneous BSE for the vertex function Γ i V i = V (0) i + K ij V j . (8) P [GeV ] g - (f( P )) PionSigmaRhoAxial -1-0.5 0 0.5 1 1.5-1 -0.5 0 0.5 1 P [GeV ] AxialRhoSigmaPion
Fig. 7.
Eigenvalue ( top ) and vertex pole ( bottom ) extrapola-tion from P > The obvious difference between this and the homogeneousBSE is the inhomogeneous term Γ (0) i . Its introductionleads to several important changes to the solution. Set-ting the relative momentum p to zero, for convenience, weobserve the appearance of poles V i ( P ) ∼ P + M , (9)as one approaches the bound-state P ∼ − M . Then, thedetermination of a bound-state mass is reduced to lookingfor zeros in 1 /V i . Typically, the leading amplitude is usedas point of reference, and one employs the method of bi-conjugate gradient (stabilised) for solution.Restricting ourselves to spacelike momenta P requiresonce more the use of fit functions and extrapolation. Here,the most useful are rational polynomials R n,m ( x ) = (cid:80) ni =0 a i x i (cid:80) i =1 ,m b i x i . (10)Note, that since the coefficients a i , b i are obtained throughleast-squares fitting, the resulting function is not a truePad´e. Regardless, the procedure appears quite reliable ascan be seen in the bottom panel of Fig. 7. ilan Vujinovic , Richard Williams : Hadronic bound-states in SU(2) from Dyson–Schwinger Equations 5 Table 1.
Results for vertex pole extrapolation for QCDrainbow-ladder in the chiral limit, compared with the resultcomputed through direct analytic continuation. All units arein MeV. The points P are taken from the region (0 , L ); theerrors on extrapolated results come from the fitting procedure. J P C calc R (2 , ( L = 0 . R (2 , ( L = 1 . − + ++
658 657(23) 656(23)1 −−
738 731(27) 728(27)1 ++
900 899(33) 899(33)We summarize our results for vertex pole approxima-tion in Table. 1. The results obtained with eigenvalue ex-trapolation are not quoted as the method performs ratherpoorly, especially in the 1 ++ channel (see top panel ofFig. 7). In either of the extrapolation techniques there aretwo principal sources of uncertainty for the mass values.One comes from the fitting procedure, since the fit func-tion coefficients ( a i , b i of Eq. (10) for vertex pole method)come with their own error bars. These errors are straight-forward to quantify, and the resulting uncertainties formeson masses are quoted in parentheses in Table. 1.A second source of errors has to do with the applica-bility of the extrapolation procedure, as one would expectthe whole method to become less reliable as one probesdeeper into the P < P ,with the total momentum sampled in the region (0 , L ) (inGeV ), and with L given in the table. In the next sec-tion we employ the method with L = 0 .
5, which appearsempirically to have the best performance.
As already highlighted, the majority of model dependencestems from the truncation of the quark-gluon vertex DSE.Other parts of the calculation are constrained either by theunderlying gauge dynamics (i.e. the ghost and gluon prop-agator which are taken from appropriate lattice or contin-uum calculations) or by chiral symmetry (in the processof truncating the BSE kernel). Thus, we can test the sen-sitivity of the truncation by varying the solution of thequark-gluon vertex within the constraints imposed by chi-ral symmetry breaking and the axWTI.The natural step is to dress the three-gluon vertex.This is motivated by both the 3PI formalism [88] and -2-1.5-1-0.5 0 0.5 1 1.5 2 2.510 -4 -3 -2 -1 Γ d r e ss i ng s [arbitrary units] Fig. 8.
Dressing for the three-gluon vertex, with s = (1 / · ( p + p + p ) and a = s = 0, see Eq. (50) of [63]. The momen-tum variable s is in arbitrary units: scale setting procedure isdescribed in Section 4.2. through the effective resummation of neglected diagramsin the full DSE for the quark-gluon vertex. This in turnenables us to give a rough estimate as to the impact ofincluding additional corrections on our results. It is suf-ficient to describe the full three-gluon vertex in Landaugauge by its tree-structure and one function of a symmet-ric variable s = (1 / · ( p + p + p ) [63] Γ µνρ ( p , p , p ) = A ( s ) · Γ (0) µνρ ( p , p , p ) . (11)The dressing function A ( s ) is obtained by solving thethree-gluon vertex DSE in a ‘ghost triangle’ approxima-tion, depicted in Fig. 9. The details of the calculationcan be found in [63]. The resultant dressing function isshown in Fig. 8. Information available from continuumnon-perturbative studies of the three-gluon vertex [63, 89–91] suggests that both the truncation of Fig. 9, and therestriction of possible tensor structures to the tree-levelterm, provide a reasonable phenomenological descriptionof this Green’s function. The effect which the dressedthree-gluon vertex has on the hadron masses can be seenin Table 2.There is one further extension to our model that ispossible, which is the inclusion of the so-called Abeliandiagram in the quark-gluon vertex DSE, see Ref. [46].This introduces no complications in the evaluation of thequark-gluon vertex itself, and through the ‘cutting’ proce-dure it is straightforward to construct a solvable BSE ker-nel which is consistent with axWTI [37]. This BSE kernelwould contain diagram with a new topology – the so-calledcrossed ladder diagram – which increases the algebraic andnumerical effort considerably. However, in previous calcu-lations the Abelian contribution has been shown to havea small effect on meson masses, typically less than twopercent [92], which would similarly apply to our presentinvestigation. For these reasons, and in light of other un-certainties of continuum and lattice investigations, we feelthat it is justified to ignore this extension for now. Milan Vujinovic , Richard Williams : Hadronic bound-states in SU(2) from Dyson–Schwinger Equations Comparison of our results with the lattice [12, 13] requiresthe scale to be set by equating the electroweak (EW) scalewith the pseudoscalar meson (‘pion’) decay constant, i.e. v EW = f π = 246 GeV. This puts the theory under investi-gation in the context of dynamical EW symmetry break-ing, otherwise known as Technicolor (TC) [93, 94].The drawbacks that the SU(2) model discussed here(and any other model with QCD-like dynamics) faces asa Technicolor template are by now well known. These in-clude the problems with precision tests on flavor-changingneutral currents [95], and the composite ‘Higgs boson’which is expected to be very heavy. This latter problemis seen here, whereupon we do not find an isoscalar scalar(‘sigma’) meson (a TC version of the Higgs boson) below1 .
33 TeV. This situation, however, might change drasti-cally if one considers explicitly the couplings to StandardModel particles [96], or more general EW embeddings [97].Another promising approach to Technicolor phenomenol-ogy is to use nearly conformal theories as Technicolor tem-plates [98–101]. As we are presently concerned with theQCD-like aspects of the model under investigation, wewill not comment on its possible Technicolor applicationsfurther.Ground state masses for various J P C mesons are shownfor both the rainbow-ladder (RL) and beyond rainbow-ladder (BRL) truncations in Table 2, where they are ad-ditionally compared with the relevant lattice calculations.RL results were obtained by means of direct analytic con-tinuation, whilst those of BRL were extrapolated from theregion of spacelike P via the inverse vertex function. Thepion decay constant, which is used to set the scale of thecalculation, is evaluated via the relation [102]: f π = Z N c √ P tr (cid:90) k Γ π ( k, − P ) S ( k + ) γ /P S ( k − ) , (12)where k ± = k ± P/ Γ π is the pion BSE amplitudenormalised according to the Nakanishi condition [103]. InQCD, the conventions employed in the above equationwould correspond to the value f π = 93 MeV. When work-ing in the region of spacelike P , the definition of Eq. (12)can be used without approximations only in the chirallimit, since the pion amplitude Γ π can be obtained forthe case P →
0. For non-chiral quarks, and thus non-vanishing pion mass, the calculation would have to be setup for complex total momentum, which is a formidabletask in a BRL setting [104].
Fig. 9.
The truncated DSE for the three-gluon vertex. Toensure that bose-symmetry is maintained the right-hand sideis averaged over all cyclic permutations. m q [arbitrary units] m π Fig. 10.
Adherence of the calculated pion mass (squared) tothe GMOR relation, as a function of the (corrected) quarkmass m q . For the discussion of results it would be useful to havean estimate on the mass of an isoscalar scalar meson,calculated in a method different from our DSE/BSE ap-proach. Since the lattice results for this particle are yet tocome, we will use the values obtained by means of grouptheory scaling, which for the model under investigationgives m σ ∈ [1 , .
5] TeV [96]. Taking this into considera-tion, it seems that the RL method fares well for the sigmameson, and to a lesser extent, the rho meson. In the 1 ++ channel, this truncation performs inadequately, with a re-sult which deviates by about 30 percent from the centrallattice value. It is arguable whether or not one can mod-ify the RL method so that it is better suited for an SU(2)theory, thus performing reasonably well for all consideredmesons. Based on our current results, and given the limi-tations of the RL framework, we are skeptical towards thisprospect.On the other hand, the results of the BRL approachcompare well with lattice, especially when employing thedressed three-gluon vertex. Since there are considerableerror margins present in both the continuum and latticeinvestigations, stronger statements about the agreementof our methods will have to wait for more refined calcula-tions.Regarding the continuum calculation, dressing of thethree-gluon vertex seems to lead to a better agreementwith the discretised approach, but the overall impact ofthis modification is relatively mild, and all meson massesare rather robust in this respect. This leaves open thepossibility that more elaborate modifications of our model(i. e. inclusion of additional diagrams and higher n -pointGreen’s functions in the quark-gluon vertex DSE) mightnot change the results appreciably. However, note thatdynamical contributions that can collectively be termed‘pion cloud’ effects are known to be important, and arethe focus of present and future investigations. ilan Vujinovic , Richard Williams : Hadronic bound-states in SU(2) from Dyson–Schwinger Equations 7 Table 2.
Chiral limit results for meson masses in rainbow-ladder (RL) and beyond rainbow-ladder (BRL) truncations, comparedwith lattice data for an SU(2) theory. All units are in TeV. Errors of the BRL results come from the extrapolation procedure.For the 0 ++ state, our continuum result is for an isoscalar; lattice results are forthcoming. J P C
RL BRL, bare 3g vertex BRL, dressed 3g vertex Lattice, from [12, 13]0 − + ++ . . −− . . . ± . ++ . . . ± . f π ) as a function of current quark mass. Bothplots correspond to a calculation with a bare three-gluonvertex. The results shown in Fig. 11 seem to compare wellwith the ones shown in Fig. 6 of [13]: however, a directcomparison is not possible since we don’t have enoughinformation to relate our m q to the ones employed in [13].As a final remark, we note that the calculation of thebaryonic spectrum in this theory does not require any ad-ditional effort. An SU(2) gauge theory possesses an en-larged (Pauli-G¨ursey) flavor symmetry, which implies thatchiral multiplets will contain both mesons and baryons(diquarks). In other words, a meson with J P quantumnumbers will be degenerate with a J − P diquark. This de-generacy (which breaks down if the chemical potential israised above some critical value µ c ) has been confirmed innumerous lattice investigations [2, 4, 5, 12, 13]. m q [arbitrary units] m/f π AxialRho
Fig. 11. J = 1 meson masses (in units of chiral limit f π ) asa function of current quark mass. Bands correspond to uncer-tainties due to the extrapolation. The right-hand side of thevertical line corresponds to the region where m ρ ≤ m π . We presented a Dyson-Schwinger/Bethe-Salpeter calcula-tion of ground state hadron masses in a theory with twocolors and two fundamentally charged Dirac fermions. Weemployed a novel beyond rainbow-ladder method and ob-tained good agreement with lattice results for spin onemesons: however, improved calculations will be needed toreduce uncertainties in both lattice and continuum ap-proaches.For J = 0 mesons, we demonstrated that chiral dy-namics are satisfied (i.e. the GMOR relation holds) andobtained the mass of the sigma meson in good agreementwith the analysis based on group theory scaling. Addition-ally, we showed that the rainbow-ladder method performsunsatisfactorily in this strongly-interacting template. Thisunderlines the need to use more sophisticated techniqueswhen studying generic non-Abelian gauge theories.Besides masses, the beyond rainbow-ladder approachwe outlined here can also be used to study hadronic de-cays and form factors. A first step towards accessing thesequantities is to extend the calculation to complex Eu-clidean momenta. However, the technical complicationswhich arise are considerable and are subject to future in-vestigation. Acknowledgments
We would like to thank R. Alkofer, C. S. Fischer, A. Maas,H. Sanchis-Alepuz, and F. Sannino for useful discussionsand a critical reading of this manuscript. This work wassupported by the Helmholtz International Center for FAIRwithin the LOEWE program of the State of Hesse, a Lise-Meitner fellowship M1333–N16 from the Austrian ScienceFund (FWF), and from the Doktoratskolleg “Hadrons inVacuum, Nuclei and Stars” (FWF) DK W1203-N16. Fur-ther support by the European Union (Hadron Physics 3project “Exciting Physics of Strong Interactions”) is ac-knowledged.
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